Path Integral Over Conformally Self-Dual Geometries

TL;DR

This paper explores a restricted path integral in four-dimensional quantum gravity focusing on conformally self-dual metrics, linking it to two-dimensional quantum gravity methods and deriving critical properties.

Contribution

It generalizes the David-Distler-Kawai approach to four dimensions, providing new insights into conformal anomalies and critical phenomena in quantum gravity.

Findings

Derivation of critical exponents for conformally self-dual metrics

Identification of an analog of the c=1 barrier in four dimensions

Connections established with Weyl and topological gravity

Abstract

The path integral of four dimensional quantum gravity is restricted to conformally self-dual metrics. It reduces to integrals over the conformal factor and over the moduli space of conformally self--dual metrics and can be studied with the methods of two dimensional quantum gravity in conformal gauge. The conformal anomaly induces an analog of the Liouville action. The proposal of David, Distler and Kawai is generalized to four dimensions. Critical exponents and the analog of the c=1 barrier of two dimensional gravity are derived. Connections with Weyl gravity and four dimensional topological gravity are suggested.